Pricing Options And Parameter Estimate Under Double Exponential Jump Diffusion Process

Since F.Black, M.Scholes and R.Merton made a major breakthrough in the pricing of financial derivatives, rapid progress has been gained in the theory and application of mathematical finance. With the deepness of study in financial practice, especially, from the serious impact concerning recent rare financial events and many questions of financial reform, etc. the Black-Scholes model based on the Brown and normal distribution is found to be not appropriate for changes of the modern financial market. In 1976, Merton established firstly a jump diffusion model where the jump risks are unsystematic and the jump magnitude of the log of the asset price is assumed to be a normal distribution, and consider option pricing of European option. Hereafter Merton's work, many research achievements have been gained. However,despite the success of the Black-Scholes and Merton model, recent empirical works indicate the inability of such two models to capture the true features of asset fluctuating, and suggest:(1) the jump risks can not be ignored, and may implicate some important economical interpretation;(2) the asset returns may represent non-asymmetric leptokurtic features and "implied volatility smile".In recent decades, many research modified Black- Scholes formula by explaining its two limitation, but the common problem is that it is difficulty to obtain a model analytical solution of option pricing for these models, at the same time, These models did not properly reflect the high peak and non-symmetrical features, especially the high peak feature.In 2002, Kou proposed double exponential jump diffusion model, the most important is that double exponential jump diffusion model can generates a highly skewed and leptokurtic distribution, in addition, DEJD leads to tractable analysis pricing formulas for European and path-dependent options. Accordingly, the double exponential representation has gained wide acceptance. In this paper, we derive the pricing formulas for European option and exotic options by using Martingale method. However, estimate and empirical assessment of this model has received little attention to date. In this paper we propose a Bayesian method to estimate the double exponential jump diffusion model. The approach is based on the Markov chain Monte Carlo(MCMC) methods with the likelihood of the discredited process as the approximate posterior likelihood. We demonstrate that the MCMC method provides a...